3D Modeling of the GRACE Satellites Positions Using and Integration of the Hermite Polynomial Approximation and the Legendre Interpolation

The Gravity Recovery And Climate Experiment (GRACE), twin satellites launched in March 2002, are making detailed measurements of the Earth's gravity field. It will yield discoveries about gravity and the Earth's natural systems.
Different sensors and instruments have been placed in the GRACE satellites to fulfill the primary scientific objective of the mission in mapping the Earth’s gravity field and its temporal variations. The K-band inter-satellite ranging system observes the key observations of the twin satellites which continuously records the changes of the inter-satellite distance. However, the two satellites three dimensional (3D) positions are recorded using the Global Positioning System (GPS) with lower sampling rate. Densification of the position vector with a sampling rate compatible with that of the K-band ranging system is the main purpose of this article.
Interpolation methods are the simplest way to calculate the position of the satellites between a few measured positions. The Lagrange interpolation method is the most frequently used scheme for orbit interpolation purposes. However, the accuracy of the method is not convincing for satellite gravimetry applications. on the other hand, the Hermite polynomials approximation can be used to combine a function and its derivatives for interpolation applications. It has shown its high performance wherever a function and its derivates have been observed.
In the GRACE mission, only 3D positions are observed by the onboard GPS receivers. Moreover, the K-band system measurement can be expressed as a nonlinear function of the relative position and velocity of the two satellites. Consequently, the Hermite polynomial approximation cannot be employed in its original form because of the nonlinearity of derivatives. Herein, we propose the idea of integration of the Lagrange interpolation and Hermite polynomials for coordinate estimation. The Lagrange interpolation is used to provide approximate coordinates between the sampling points. Finally, the Hermite polynomial approximation is utilized for simultaneous adjustment of all the GPS-derived positions, K-band measurements and the approximate position derived from the Lagrange interpolation. Numerical analysis shows that the proposed method outperforms both the Lagrange interpolation and the Hermite polynomial approximation in terms of accuracy.